Mathematical Sciences: Reciprocity for Fredholm Operators

算子论是现代分析的中心学科。它起源于二十世纪初对数学物理和偏微分方程式的研究。当时，人们可以看到，平衡论、振动、量子理论等中的许多物理问题可以通过模拟这些现象的积分方程组进行富有成效的研究。因此，从希尔伯特、冯·诺伊曼和其他巨人丰富的头脑中，算子理论的主题已经成长为此类研究的中心位置，在核心数学中也是如此。这种方法的核心是深入研究操作员的频谱。对于自伴或正规算子，这一理论现在是整个分析的标准技术，而谱定理为所有这类算子提供了必要的基础。因此，当前的前沿是研究非正规算子的结构，例如在弗雷德霍尔姆理论中。在这一前沿，凯里教授和他的合作者平卡斯教授是世界领袖，在深入、概念创新、基础和跨学科的工作方面享有盛誉。他们的主函数理论定义在某些算子的谱上，是普通指数的广泛推广。这一新的研究合作在函数代数的研究中建立了主函数和与遍历流有关的函数的零点分布、次正规算子的结构和闭流理论之间的联系。这项工作特别丰富，因为它直接将算子理论的思想与数学的其他分支结合起来，特别是交换环理论和代数几何。目前，在算子理论中的其他技术应用中，还包括交换Toeplitz算子、线性和非线性偏微分算子的应用。

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. Thus, from the fertile minds of Hilbert, von Neumann, and other giants the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator. For self-adjoint or normal operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. The current frontier, therefore, is the study of the structure of non-normal operators, as for example in the Fredholm theory. At this frontier, Professor Carey and his collaborator, Professor Pincus, are world leaders with established reputations for deep, conceptually innovative, fundamental, interdisciplinary work. Their theory of the principal function, defined on the spectrum of certain operators, is a broad extension of the ordinary index. This new research collaboration has established connections between the principal function and the distribution of zeroes of functions associated with ergodic flows, the structure of subnormal operators and the theory of closed currents in the study of function algebras. This work is particularly fertile as it directly combines operator theoretic ideas with other branches of mathematics, notably commutative ring theory and algebraic geometry. Currently, applications to the theory of commuting Toeplitz operators and linear and nonlinear partial differential operators are envisaged, among other technical applications in operator theory.